Brillouin-zone databases

Reciprocal-space group

The k-vectors are vectors of the reciprocal space of G and can be represented as

where {a₁^*, a₂^*, a₃^*} is the basis of the reciprocal lattice L^*. The basis {a₁^*, a₂^*, a₃^*} is the dual basis of {a₁, a₂, a₃} of L (the vector lattice of G) and its vectors a_i^* are defined by the relations a_i⋅a_j^* = 2πδ_ij where δ_ij is the Kronecker symbol. The symmetry properties of the wave vectors are described by the so-called reciprocal-space groups (G)^* whose elements are operations of the type (W, K) = (I, K) (W, o) with W ∈ Ḡ and K ∈ L^*
(I is the (3x3) unit matrix, o is the (3x1) zero column, and Ḡ is the point group of the space group G). The group (G)^* is the semidirect product of the point group Ḡ and the translation group of the reciprocal lattice L^* of G.

From its definition it follows that the reciprocal-space group (G)^* is isomorphic to a symmorphic space group G₀ (for symmorphic space groups, cf. ITA, Section 8.1.6). Space groups of the same type define the same type of reciprocal-space group (G)^*. In addition, as (G)^* does not depend on the column parts of the space-group operations, all space groups of the same arithmetic crystal class determine the same type of (G)^* (Wintgen, 1941) (for definition and symbols of arithmetic crystal classes, see Section 8.2.3 of ITA). For about 2/3 of the space groups, (G)^* and G belong to the same arithmetic crystal class, i. e given the space group G, its reciprocal-space group (G)^* is isomorphic to the symmorphic group G₀ related to G. However, there are a number of cases when the arithmetic crystal classes of G and (G)^* are different. For example, if the lattice symbol of G is F or I, then the lattice symbol of (G)^* is I or F. (The tetragonal space groups form an exception from this rule; for these the symbol I persists). The rest of the exceptions are listed in the table:

Uni-arm description

Two k-vectors of a Wyckoff position are called uni-arm if one can be obtained from the other by parameter variation. The description of k-vectors stars of a Wyckoff position is called uni-arm if the k-vectors representing these stars are uni-arm. Frequently, in order to achieve uni-arm description, it is necessary to transform k-vectors to equivalent ones. In addition, to enable a uni-arm description, symmetry lines outside the asymmetric unit may be selected as orbit representatives. Such a segment of a line is called a flagpole. In analogy to the flagpoles, symmetry planes outside the asymmetric unit may be selected as orbit representatives. Such a segment of a plane is called a wing.

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